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QwQ: Reflect Deeply on the Boundaries of the Unknown

November 28, 2024 * 22 min * 4496 words * Qwen Team | Translations:

  * Jian Ti Zhong Wen 

GITHUB HUGGING FACE MODELSCOPE DEMO DISCORD

Note: This is the pronunciation of QwQ: /kwju:/ , similar to the word
"quill".

What does it mean to think, to question, to understand? These are the
deep waters that QwQ (Qwen with Questions) wades into. Like an
eternal student of wisdom, it approaches every problem - be it
mathematics, code, or knowledge of our world - with genuine wonder
and doubt. QwQ embodies that ancient philosophical spirit: it knows
that it knows nothing, and that's precisely what drives its
curiosity. Before settling on any answer, it turns inward,
questioning its own assumptions, exploring different paths of
thought, always seeking deeper truth. Yet, like all seekers of
wisdom, QwQ has its limitations. This version is but an early step on
a longer journey - a student still learning to walk the path of
reasoning. Its thoughts sometimes wander, its answers aren't always
complete, and its wisdom is still growing. But isn't that the beauty
of true learning? To be both capable and humble, knowledgeable yet
always questioning? We invite you to explore alongside QwQ, embracing
both its insights and its imperfections as part of the endless quest
for understanding.

Limitations#

QwQ-32B-Preview is an experimental research model developed by the
Qwen Team, focused on advancing AI reasoning capabilities. As a
preview release, it demonstrates promising analytical abilities while
having several important limitations:

 1. Language Mixing and Code-Switching: The model may mix languages
    or switch between them unexpectedly, affecting response clarity.
 2. Recursive Reasoning Loops: The model may enter circular reasoning
    patterns, leading to lengthy responses without a conclusive
    answer.
 3. Safety and Ethical Considerations: The model requires enhanced
    safety measures to ensure reliable and secure performance, and
    users should exercise caution when deploying it.
 4. Performance and Benchmark Limitations: The model excels in math
    and coding but has room for improvement in other areas, such as
    common sense reasoning and nuanced language understanding.

Performance#

Through deep exploration and countless trials, we discovered
something profound: when given time to ponder, to question, and to
reflect, the model's understanding of mathematics and programming
blossoms like a flower opening to the sun. Just as a student grows
wiser by carefully examining their work and learning from mistakes,
our model achieves deeper insight through patient, thoughtful
analysis. This process of careful reflection and self-questioning
leads to remarkable breakthroughs in solving complex problems. Our
journey of discovery revealed the model's exceptional ability to
tackle some of the most challenging problems in mathematics and
programming, including:

  * GPQA: A Graduate-Level Google-Proof Q&A Benchmark, a challenging
    benchmark for evaluating scientific problem-solving abilities
    through grade school level questions.
  * AIME: American Invitation Mathematics Evaluation, which tests
    mathematical problem solving with arithmetic, algebra, counting,
    geometry, number theory, and probability and other secondary
    school math topics.
  * MATH-500: The 500 test cases of the MATH benchmark, a
    comprehensive dataset testing mathematical problem-solving.
  * LiveCodeBench: A challenging benchmark for evaluating code
    generation and problem solving abilities in real-world
    programming scenarios.

[QwQ-32B-Preview_result][QwQ-32B-Preview_acc_with_k]

Specifically, QwQ demonstrates remarkable performance across these
benchmarks, achieving impressive scores of 65.2% on GPQA, showcasing
its graduate-level scientific reasoning capabilities; 50.0% on AIME,
highlighting its strong mathematical problem-solving skills; 90.6% on
MATH-500, demonstrating exceptional mathematical comprehension across
diverse topics; and 50.0% on LiveCodeBench, validating its robust
programming abilities in real-world scenarios. These results
underscore QwQ's significant advancement in analytical and
problem-solving capabilities, particularly in technical domains
requiring deep reasoning.

Demo Cases#

In the following examples, we invite you to witness the profound
depths of QwQ-32B-Preview's contemplative abilities. Like a seeker of
wisdom on an endless journey of discovery, the model demonstrates its
capacity for deep introspection - questioning its own assumptions,
engaging in thoughtful self-dialogue, and carefully examining each
step of its reasoning process. Through these cases, you'll observe
how QwQ embodies the timeless understanding that true knowledge
emerges not from hasty conclusions, but from patient inquiry and the
courage to challenge one's own certainties in the eternal pursuit of
truth.

Example: Logical Reasoning Next
User
Please add a pair of parentheses to the incorrect equation: 1 + 2 * 3
+ 4 * 5 + 6 * 7 + 8 * 9 = 479, to make the equation true.
QwQ-32B-Preview

Let's tackle this problem step by step. The equation given is:

1 + 2 * 3 + 4 * 5 + 6 * 7 + 8 * 9 = 479

But it's incorrect, and I need to add a pair of parentheses to make
it true. Okay, first things first, I should understand what the
current value of the left side is without any parentheses.

Let's recall the order of operations: PEMDAS (Parentheses, Exponents,
Multiplication and Division, Addition and Subtraction). So,
multiplication comes before addition.

So, calculating step by step:

2 * 3 = 6

4 * 5 = 20

6 * 7 = 42

8 * 9 = 72

Now, plugging these back into the equation:

1 + 6 + 20 + 42 + 72

Adding these up:

1 + 6 = 7

7 + 20 = 27

27 + 42 = 69

69 + 72 = 141

So, without any parentheses, the left side equals 141, but it should
be 479. That's quite a difference. So, I need to add a pair of
parentheses to make it equal to 479.

I need to think about where to place the parentheses to alter the
order of operations to achieve the desired result.

Let me consider different possibilities:

 1. Parenthesizing individual terms:

For example, (1 + 2) * 3 + 4 * 5 + 6 * 7 + 8 * 9

Calculate inside the parentheses first:

(1 + 2) = 3

Then, 3 * 3 = 9

Now, the equation becomes:

9 + 4 * 5 + 6 * 7 + 8 * 9

Following order of operations:

4 * 5 = 20

6 * 7 = 42

8 * 9 = 72

So, 9 + 20 + 42 + 72 = 143

Still not 479.

 2. Parenthesizing more terms:

What if I do (1 + 2 * 3) + 4 * 5 + 6 * 7 + 8 * 9

First, calculate inside the parentheses:

1 + 2 * 3

Here, multiplication comes before addition, so 2 * 3 = 6, then 1 + 6
= 7

So, the equation becomes:

7 + 4 * 5 + 6 * 7 + 8 * 9

Again, following order of operations:

4 * 5 = 20

6 * 7 = 42

8 * 9 = 72

So, 7 + 20 + 42 + 72 = 141

Same as before.

 3. Let's try (1 + 2) * (3 + 4) * 5 + 6 * 7 + 8 * 9

First, calculate inside the parentheses:

(1 + 2) = 3

(3 + 4) = 7

So, 3 * 7 * 5 = 105

Now, the equation becomes:

105 + 6 * 7 + 8 * 9

Following order of operations:

6 * 7 = 42

8 * 9 = 72

So, 105 + 42 + 72 = 219

Still not 479.

 4. Maybe trying to group more terms:

(1 + 2 * 3 + 4) * 5 + 6 * 7 + 8 * 9

First, calculate inside the parentheses:

1 + 2 * 3 + 4

Multiplication first: 2 * 3 = 6

Then, 1 + 6 + 4 = 11

So, 11 * 5 = 55

Now, the equation becomes:

55 + 6 * 7 + 8 * 9

Following order of operations:

6 * 7 = 42

8 * 9 = 72

So, 55 + 42 + 72 = 169

Getting closer, but still not 479.

 5. Let's try a different approach. Maybe grouping the additions
    together:

1 + (2 * 3 + 4 * 5 + 6 * 7 + 8 * 9)

First, calculate inside the parentheses:

2 * 3 = 6

4 * 5 = 20

6 * 7 = 42

8 * 9 = 72

So, 6 + 20 + 42 + 72 = 140

Then, 1 + 140 = 141

Same as before.

 6. Perhaps grouping two multiplications together:

1 + 2 * (3 + 4) * 5 + 6 * 7 + 8 * 9

First, calculate inside the parentheses:

(3 + 4) = 7

So, 2 * 7 * 5 = 70

Now, the equation becomes:

1 + 70 + 6 * 7 + 8 * 9

Following order of operations:

6 * 7 = 42

8 * 9 = 72

So, 1 + 70 + 42 + 72 = 185

Still not 479.

 7. Maybe grouping more multiplications:

1 + 2 * 3 + 4 * (5 + 6) * 7 + 8 * 9

First, calculate inside the parentheses:

(5 + 6) = 11

So, 4 * 11 * 7 = 308

Now, the equation becomes:

1 + 2 * 3 + 308 + 8 * 9

Following order of operations:

2 * 3 = 6

8 * 9 = 72

So, 1 + 6 + 308 + 72 = 387

Closer, but still not 479.

 8. Let's try grouping the last terms:

1 + 2 * 3 + 4 * 5 + 6 * (7 + 8) * 9

First, calculate inside the parentheses:

(7 + 8) = 15

So, 6 * 15 * 9 = 810

Now, the equation becomes:

1 + 2 * 3 + 4 * 5 + 810

Following order of operations:

2 * 3 = 6

4 * 5 = 20

So, 1 + 6 + 20 + 810 = 837

Now that's way over 479.

Maybe I need to group terms in a way that subtraction is implied, but
since there are no subtractions in the original equation, perhaps I
need to consider that.

Wait, maybe the problem allows for adding parentheses to change the
operations indirectly.

Alternatively, perhaps the parentheses need to group additions in a
specific way.

Let me try grouping the first few terms:

(1 + 2 * 3 + 4 * 5) + 6 * 7 + 8 * 9

First, calculate inside the parentheses:

2 * 3 = 6

4 * 5 = 20

So, 1 + 6 + 20 = 27

Now, the equation becomes:

27 + 6 * 7 + 8 * 9

Following order of operations:

6 * 7 = 42

8 * 9 = 72

So, 27 + 42 + 72 = 141

Same as before.

 9. Maybe grouping further:

(1 + 2 * 3 + 4 * 5 + 6 * 7) + 8 * 9

First, calculate inside the parentheses:

2 * 3 = 6

4 * 5 = 20

6 * 7 = 42

So, 1 + 6 + 20 + 42 = 69

Now, the equation becomes:

69 + 8 * 9

Following order of operations:

8 * 9 = 72

So, 69 + 72 = 141

Still not 479.

10. Let's try grouping differently:

1 + 2 * (3 + 4 * 5 + 6 * 7 + 8 * 9)

First, calculate inside the parentheses:

3 + 4 * 5 + 6 * 7 + 8 * 9

Following order of operations inside:

4 * 5 = 20

6 * 7 = 42

8 * 9 = 72

So, 3 + 20 + 42 + 72 = 137

Now, 2 * 137 = 274

Then, the equation becomes:

1 + 274 = 275

Still not 479.

11. Maybe trying to group the additions and multiplications in a
    different way:

1 + (2 * 3 + 4 * 5 + 6 * 7 + 8 * 9)

Wait, I did this earlier and got 141.

Hmm.

Let me think differently. Maybe I need to group terms to create a
larger multiplication.

For example:

1 + 2 * (3 + 4 * 5 + 6 * 7 + 8 * 9)

First, calculate inside the parentheses:

3 + 4 * 5 + 6 * 7 + 8 * 9

Following order of operations:

4 * 5 = 20

6 * 7 = 42

8 * 9 = 72

So, 3 + 20 + 42 + 72 = 137

Now, 2 * 137 = 274

Then, 1 + 274 = 275

Still not 479.

12. Let's try another approach:

(1 + 2) * (3 + 4) * (5 + 6) * (7 + 8) * 9

Wait, that's adding multiple sets of parentheses, which is not
allowed since I can only add one pair.

Back to the original problem: only one pair of parentheses can be
added.

Maybe I need to group a subtraction or division, but those operations
aren't present in the original equation.

Wait, perhaps I need to consider that adding parentheses can create a
negative term, but again, there are no subtractions in the original
equation.

Alternatively, maybe the parentheses are meant to group additions in
a specific way to create a larger number.

Let me try grouping more terms together:

1 + 2 * (3 + 4 * 5 + 6 * 7 + 8 * 9)

First, inside the parentheses:

3 + 4 * 5 + 6 * 7 + 8 * 9

Following order of operations:

4 * 5 = 20

6 * 7 = 42

8 * 9 = 72

So, 3 + 20 + 42 + 72 = 137

Now, 2 * 137 = 274

Then, 1 + 274 = 275

Same as before.

13. Maybe trying to group the last few terms:

1 + 2 * 3 + 4 * 5 + 6 * (7 + 8 * 9)

First, inside the parentheses:

7 + 8 * 9

Following order of operations:

8 * 9 = 72

Then, 7 + 72 = 79

Now, 6 * 79 = 474

So, the equation becomes:

1 + 2 * 3 + 4 * 5 + 474

Following order of operations:

2 * 3 = 6

4 * 5 = 20

So, 1 + 6 + 20 + 474 = 501

Close to 479, but still not there.

14. Maybe adjusting the grouping slightly:

1 + 2 * 3 + 4 * (5 + 6 * 7 + 8 * 9)

First, inside the parentheses:

5 + 6 * 7 + 8 * 9

Following order of operations:

6 * 7 = 42

8 * 9 = 72

So, 5 + 42 + 72 = 119

Now, 4 * 119 = 476

Then, the equation becomes:

1 + 2 * 3 + 476

Following order of operations:

2 * 3 = 6

So, 1 + 6 + 476 = 483

Still not 479.

15. Let's try grouping the first part differently:

(1 + 2 * 3 + 4) * (5 + 6 * 7 + 8 * 9)

First, calculate inside the first set of parentheses:

1 + 2 * 3 + 4

Following order of operations:

2 * 3 = 6

Then, 1 + 6 + 4 = 11

Now, the second set of parentheses:

5 + 6 * 7 + 8 * 9

Following order of operations:

6 * 7 = 42

8 * 9 = 72

So, 5 + 42 + 72 = 119

Now, 11 * 119 = 1309

Way over 479.

16. Maybe trying to group only two terms:

1 + 2 * (3 + 4) * 5 + 6 * 7 + 8 * 9

Wait, I did this earlier and got 185.

Hmm.

Let me think differently. Maybe I need to group terms to create a
division or subtraction, but since those operations aren't present,
perhaps I need to consider that the parentheses can be used to create
a fraction or some other operation, but that might be reaching too
far.

Alternatively, maybe the problem allows for adding a pair of
parentheses around the entire equation, but that doesn't make sense.

Wait, perhaps the parentheses are meant to group additions in a
specific way to create a larger number.

Let me try grouping more terms together for multiplication.

For example:

1 + 2 * (3 + 4 * 5 + 6) * 7 + 8 * 9

First, inside the parentheses:

3 + 4 * 5 + 6

Following order of operations:

4 * 5 = 20

So, 3 + 20 + 6 = 29

Now, 2 * 29 * 7 = 406

Then, the equation becomes:

1 + 406 + 8 * 9

Following order of operations:

8 * 9 = 72

So, 1 + 406 + 72 = 479

Wait a minute, that's exactly 479!

Let me double-check:

First, inside the parentheses: 3 + 4 * 5 + 6

4 * 5 = 20

Then, 3 + 20 + 6 = 29

Then, 2 * 29 * 7 = 2 * 203 = 406

Then, 1 + 406 = 407

Plus 8 * 9 = 72

So, 407 + 72 = 479

Perfect! So, by adding parentheses around (3 + 4 * 5 + 6), and
multiplying that sum by 2 and 7, we get the desired result.

So, the corrected equation is:

1 + 2 * (3 + 4 * 5 + 6) * 7 + 8 * 9 = 479

Final Answer

$$ \boxed{1 + 2 \times (3 + 4 \times 5 + 6) \times 7 + 8 \times 9 =
479} $$

Example: Math Next
User
Find the least odd prime factor of $2019^8+1$.
QwQ-32B-Preview

So I need to find the least odd prime factor of $ 2019^8 + 1 $. Hmm,
that seems pretty big, but I think I can break it down step by step.

First, I know that finding prime factors of large exponents can be
tricky, but there might be some properties or theorems that can help
simplify the process. Since the expression is $ 2019^8 + 1 $, maybe I
can look for patterns or ways to factor this expression.

I recall that expressions of the form $ a^n + b^n $ can sometimes be
factored, especially when $ n $ is a power of 2. In this case, $ n =
8 $, which is $ 2^3 $, so that might be useful.

Let me see if I can factor $ 2019^8 + 1 $. I know that $ a^4 + 1 $
can be factored as $ (a^4 + 1) = (a^2 + \sqrt{2}a + 1)(a^2 - \sqrt{2}
a + 1) $, but that involves irrational numbers, which isn't helpful
for factoring into integers. Maybe there's another way.

Wait, perhaps I can write $ 2019^8 + 1 $ as $ (2019^4)^2 + 1^2 $,
which resembles the sum of squares. I know that over the integers,
sums of squares don't factor nicely, but maybe there's a different
approach.

Another thought: maybe I can consider smaller exponents first and see
if there's a pattern or a recurring factor that appears.

Let's try computing $ 2019^2 + 1 $, $ 2019^4 + 1 $, and so on, and
see if I can find any common factors or patterns.

First, calculate $ 2019^2 + 1 $:

$ 2019^2 = 2019 \times 2019 $. I don't need the exact value right
now; maybe I can just keep it as $ 2019^2 $ for now.

So, $ 2019^2 + 1 = 2019^2 + 1 $. Doesn't seem immediately helpful.

Next, $ 2019^4 + 1 $. Again, it's similar to the previous expression,
and I don't see an obvious factorization.

Maybe I should try to see if there are any small prime numbers that
divide $ 2019^8 + 1 $. Since I'm looking for the least odd prime
factor, I can start testing small primes like 3, 5, 7, 11, and so on,
to see if any of them divide $ 2019^8 + 1 $.

Let's start with 3. To check if 3 divides $ 2019^8 + 1 $, I can
compute $ 2019^8 + 1 $ modulo 3.

First, find $ 2019 $ modulo 3. Since $ 2019 = 3 \times 673 $, $ 2019
\equiv 0 \pmod{3} $. Therefore, $ 2019^8 \equiv 0^8 \equiv 0 \pmod{3}
$, and $ 2019^8 + 1 \equiv 0 + 1 \equiv 1 \pmod{3} $. So, 3 does not
divide $ 2019^8 + 1 $.

Next, check 5. Compute $ 2019 $ modulo 5. $ 2019 \div 5 = 403 $ with
a remainder of 4, so $ 2019 \equiv 4 \pmod{5} $. Then, $ 2019^8 \
equiv 4^8 \pmod{5} $. Since $ 4 \equiv -1 \pmod{5} $, $ 4^8 \equiv
(-1)^8 \equiv 1 \pmod{5} $. Therefore, $ 2019^8 + 1 \equiv 1 + 1 \
equiv 2 \pmod{5} $, so 5 does not divide $ 2019^8 + 1 $.

Now, check 7. Find $ 2019 $ modulo 7. $ 2019 \div 7 = 288 $ with a
remainder of 3, so $ 2019 \equiv 3 \pmod{7} $. Then, $ 2019^8 \equiv
3^8 \pmod{7} $. Let's compute $ 3^8 $ modulo 7.

First, find the pattern of powers of 3 modulo 7:

$ 3^1 \equiv 3 \pmod{7} $

$ 3^2 \equiv 9 \equiv 2 \pmod{7} $

$ 3^3 \equiv 6 \pmod{7} $

$ 3^4 \equiv 18 \equiv 4 \pmod{7} $

$ 3^5 \equiv 12 \equiv 5 \pmod{7} $

$ 3^6 \equiv 15 \equiv 1 \pmod{7} $

Since $ 3^6 \equiv 1 \pmod{7} $, then $ 3^8 = 3^{6+2} = 3^6 \times 3^
2 \equiv 1 \times 2 \equiv 2 \pmod{7} $. Therefore, $ 2019^8 + 1 \
equiv 2 + 1 \equiv 3 \pmod{7} $, so 7 does not divide $ 2019^8 + 1 $.

Next, check 11. Compute $ 2019 $ modulo 11. $ 2019 \div 11 = 183 $
with a remainder of 6, so $ 2019 \equiv 6 \pmod{11} $. Then, $ 2019^8
\equiv 6^8 \pmod{11} $. Let's find a pattern for powers of 6 modulo
11:

$ 6^1 \equiv 6 \pmod{11} $

$ 6^2 \equiv 36 \equiv 3 \pmod{11} $

$ 6^3 \equiv 18 \equiv 7 \pmod{11} $

$ 6^4 \equiv 42 \equiv 9 \pmod{11} $

$ 6^5 \equiv 54 \equiv 10 \pmod{11} $

$ 6^6 \equiv 60 \equiv 5 \pmod{11} $

$ 6^7 \equiv 30 \equiv 8 \pmod{11} $

$ 6^8 \equiv 48 \equiv 4 \pmod{11} $

Therefore, $ 2019^8 + 1 \equiv 4 + 1 \equiv 5 \pmod{11} $, so 11 does
not divide $ 2019^8 + 1 $.

Continuing this process seems tedious. Maybe there's a better way to
approach this problem.

Let me think differently. Perhaps I can consider the expression $
2019^8 + 1 $ and see if it can be expressed in a form that reveals
its prime factors.

I recall that for expressions like $ a^4 + 4b^4 $, there's a Sophie
Germain identity: $ a^4 + 4b^4 = (a^2 + 2ab + 2b^2)(a^2 - 2ab + 2b^2)
$. Maybe I can manipulate $ 2019^8 + 1 $ into a similar form.

Wait, $ 2019^8 + 1 = (2019^4)^2 + 1^2 $. This resembles the sum of
squares, but over integers, sums of squares don't factor nicely.
Maybe I can consider complex numbers or some other algebraic
identities, but that might not help in finding integer prime factors.

Alternatively, perhaps I can consider modulo properties to find
possible prime factors. For example, if $ p $ is an odd prime that
divides $ 2019^8 + 1 $, then $ 2019^8 \equiv -1 \pmod{p} $, which
implies $ 2019^{16} \equiv 1 \pmod{p} $. This means that the order of
2019 modulo $ p $ divides 16, and since $ 2019^8 \equiv -1 \pmod{p}
$, the order can't divide 8. Therefore, the order must be 16.

By Fermat's Little Theorem, the order of 2019 modulo $ p $ divides $
p - 1 $. Therefore, $ p - 1 $ must be divisible by 16. So, $ p \equiv
1 \pmod{16} $.

This narrows down the possible primes $ p $ that could divide $ 2019^
8 + 1 $ to those primes congruent to 1 modulo 16.

Let me list some primes that are 1 modulo 16: 17, 97, 113, 193, 241,
257, etc.

Now, I can test these primes to see if any of them divide $ 2019^8 +
1 $.

Start with 17. Compute $ 2019 $ modulo 17.

First, find $ 2019 \div 17 $. 17 times 118 is 2006, so $ 2019 - 2006
= 13 $. Therefore, $ 2019 \equiv 13 \pmod{17} $.

Then, $ 2019^8 \equiv 13^8 \pmod{17} $.

Compute $ 13^2 \equiv 169 \equiv 16 \pmod{17} $

$ 13^4 = (13^2)^2 \equiv 16^2 \equiv 256 \equiv 1 \pmod{17} $

$ 13^8 = (13^4)^2 \equiv 1^2 \equiv 1 \pmod{17} $

Therefore, $ 2019^8 + 1 \equiv 1 + 1 \equiv 2 \pmod{17} $, so 17 does
not divide $ 2019^8 + 1 $.

Next, check 97. Compute $ 2019 $ modulo 97.

First, find how many times 97 fits into 2019. 97 times 20 is 1940,
and 2019 - 1940 = 79. So, $ 2019 \equiv 79 \pmod{97} $.

Then, $ 2019^8 \equiv 79^8 \pmod{97} $.

This seems a bit cumbersome to compute directly. Maybe I can find a
pattern or use Fermat's Little Theorem, which states that $ a^{p-1} \
equiv 1 \pmod{p} $ if $ p $ is prime and $ p $ does not divide $ a $.

Here, $ p = 97 $, so $ 79^{96} \equiv 1 \pmod{97} $. But I need $ 79^
8 $, which is not directly helpful. Maybe I can compute $ 79^2 $,
then $ 79^4 $, and so on.

First, compute $ 79^2 \pmod{97} $:

$ 79^2 = 6241 $. Now, divide 6241 by 97.

97 times 64 is 6208, so 6241 - 6208 = 33. Therefore, $ 79^2 \equiv 33
\pmod{97} $.

Next, $ 79^4 = (79^2)^2 \equiv 33^2 \pmod{97} $.

$ 33^2 = 1089 $. 97 times 11 is 1067, so 1089 - 1067 = 22. Therefore,
$ 79^4 \equiv 22 \pmod{97} $.

Then, $ 79^8 = (79^4)^2 \equiv 22^2 \pmod{97} $.

$ 22^2 = 484 $. 97 times 4 is 388, so 484 - 388 = 96. Therefore, $ 79
^8 \equiv 96 \pmod{97} $.

Thus, $ 2019^8 + 1 \equiv 96 + 1 \equiv 97 \equiv 0 \pmod{97} $.

So, 97 divides $ 2019^8 + 1 $. Since 97 is the smallest prime
congruent to 1 modulo 16 that we've checked and it divides $ 2019^8 +
1 $, it must be the least odd prime factor of $ 2019^8 + 1 $.

Therefore, the least odd prime factor is 97.

Final Answer

$$ \boxed{97} $$

Reflections on the Journey Ahead#

The path to understanding reasoning in large language models branches
into many directions, each offering unique insights into the nature
of machine intelligence. Our research spans multiple domains - from
process reward models that shape learning patterns, to LLM critique
that fosters deeper analysis, to multi-step reasoning that builds
complex thought, to reinforcement learning with system feedback that
enables growth through real-world interactions. While the destination
remains unclear, each step brings us closer to understanding how
machines think and reason. We don't know precisely where this journey
leads, but we continue forward with unwavering determination - toward
truth, toward intelligence, toward the realm where amazing happens.

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